Forced Vibrations of Electromagnetic Systems. 391 



To verify, the uniform field of impressed force of intensity 

 /], by elementary principles, produces the external electric 

 potential 3 



n=f 1 cos0-~, 



whose derivatives, radial and tangential, taken negatively, 

 are (151) and (147). The corresponding internal potential 



O = ^-/l r cos 0. 



But its slope does not give the force E left behind within the 

 sphere, because this E is the force of the flux. Any other 

 distribution of impressed force, with the same vorticity, will 

 lead to the same E. Our equation (135) and its companion 

 for F, derived from (134) by using (136), lead to the steady 

 field (residual) 



E = -f/ lS in0, F=t/ lC os0, . . . (152) 



the components of the true force of the flux. Add e to the 

 slope of fl to produce E *. 



F is always zero at the front of the primary wave outward, 

 and B = ^ uH. At the front of the primary wave inward F is 

 also zero, and E = — fi vE. After reflexion, F at the front 

 of the reflected wave is still zero, but now E=/4 vH. 



The electric energy ITx set up is the volume-integral of the 

 scalar product £ eD. That is, 



Ui-a/iXs^X -3-- -9-^ • • • (!53; 



But the total work done by e is 2Ui, by the general law 

 that the whole work done by impressed forces suddenly 

 started exceeds the amount representing the waste by Joule 

 heating at the final rate (when there is any), supposed to 

 start at once, by twice the excess of the electric over the 

 magnetic energy of the steady field set up. It is clear, then, 

 that when the travelling shell has gone a good way out, and 

 it has become nearly equivalent to a plane wave, its electric 

 and magnetic energies are nearly equal, and each nearly ^Ui 



* Sometimes the flux is apparently wrongly directed. For example, a 

 uniform field of impressed force from left to right in all space except a 

 spherical portion produces a flux from right to left in that portion. This 

 is made intelligible by the above. Let the impressed force act in the 

 space between r = a and r=b, a being small and b great. In the inner 

 sphere the first effects are those due to the r — a vorticity, and the flux 

 left behind is against the force. But after a time comes the wave from 

 the r—b vorticity, which sets matters right. The same applies in the 

 case of conductors, when, in fact, a long time might have to elapse before 

 the second and real permanent state conquered the first one. 

 2D2 



